Question: Simplify the following expression and state the condition under which the simplification is valid. $n = \dfrac{9r^2 + 54r + 45}{-5r^3 - 70r^2 - 225r}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ n = \dfrac {9(r^2 + 6r + 5)} {-5r(r^2 + 14r + 45)} $ $ n = -\dfrac{9}{5r} \cdot \dfrac{r^2 + 6r + 5}{r^2 + 14r + 45} $ Next factor the numerator and denominator. $ n = - \dfrac{9}{5r} \cdot \dfrac{(r + 5)(r + 1)}{(r + 5)(r + 9)}$ Assuming $r \neq -5$ , we can cancel the $r + 5$ $ n = - \dfrac{9}{5r} \cdot \dfrac{r + 1}{r + 9}$ Therefore: $ n = \dfrac{ -9(r + 1)}{ 5r(r + 9)}$, $r \neq -5$